CN110007681B - Optimization method for realizing spin stability and unfolding of rope formation by using continuous propeller - Google Patents
Optimization method for realizing spin stability and unfolding of rope formation by using continuous propeller Download PDFInfo
- Publication number
- CN110007681B CN110007681B CN201811432334.0A CN201811432334A CN110007681B CN 110007681 B CN110007681 B CN 110007681B CN 201811432334 A CN201811432334 A CN 201811432334A CN 110007681 B CN110007681 B CN 110007681B
- Authority
- CN
- China
- Prior art keywords
- formation system
- spin
- equation
- stable
- rope formation
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
- 230000015572 biosynthetic process Effects 0.000 title claims abstract description 128
- 238000000034 method Methods 0.000 title claims abstract description 68
- 238000005457 optimization Methods 0.000 title claims abstract description 14
- 238000009987 spinning Methods 0.000 claims abstract description 19
- 238000013507 mapping Methods 0.000 claims abstract description 16
- 230000008569 process Effects 0.000 claims abstract description 14
- 230000033001 locomotion Effects 0.000 claims abstract description 12
- 238000001228 spectrum Methods 0.000 claims abstract description 12
- 238000004364 calculation method Methods 0.000 claims abstract description 10
- 230000008859 change Effects 0.000 claims description 8
- 238000005381 potential energy Methods 0.000 claims description 8
- 230000005484 gravity Effects 0.000 claims description 5
- 239000011159 matrix material Substances 0.000 claims description 5
- 238000013461 design Methods 0.000 claims description 3
- 230000004069 differentiation Effects 0.000 claims description 3
- NAWXUBYGYWOOIX-SFHVURJKSA-N (2s)-2-[[4-[2-(2,4-diaminoquinazolin-6-yl)ethyl]benzoyl]amino]-4-methylidenepentanedioic acid Chemical compound C1=CC2=NC(N)=NC(N)=C2C=C1CCC1=CC=C(C(=O)N[C@@H](CC(=C)C(O)=O)C(O)=O)C=C1 NAWXUBYGYWOOIX-SFHVURJKSA-N 0.000 claims description 2
- 230000003190 augmentative effect Effects 0.000 claims description 2
- 238000012544 monitoring process Methods 0.000 claims description 2
- 230000009897 systematic effect Effects 0.000 claims description 2
- 230000009466 transformation Effects 0.000 claims description 2
- 238000002407 reforming Methods 0.000 claims 1
- 238000005755 formation reaction Methods 0.000 description 83
- 230000010355 oscillation Effects 0.000 description 13
- 230000035945 sensitivity Effects 0.000 description 7
- 238000010586 diagram Methods 0.000 description 6
- 230000007423 decrease Effects 0.000 description 4
- 230000008878 coupling Effects 0.000 description 3
- 238000010168 coupling process Methods 0.000 description 3
- 238000005859 coupling reaction Methods 0.000 description 3
- 241000282414 Homo sapiens Species 0.000 description 2
- 238000013459 approach Methods 0.000 description 2
- 230000008901 benefit Effects 0.000 description 2
- 230000003247 decreasing effect Effects 0.000 description 2
- 238000011161 development Methods 0.000 description 2
- 239000000446 fuel Substances 0.000 description 2
- 238000004088 simulation Methods 0.000 description 2
- 230000001133 acceleration Effects 0.000 description 1
- 230000009471 action Effects 0.000 description 1
- 238000004422 calculation algorithm Methods 0.000 description 1
- 230000007547 defect Effects 0.000 description 1
- 230000006872 improvement Effects 0.000 description 1
- 230000004048 modification Effects 0.000 description 1
- 238000012986 modification Methods 0.000 description 1
- 239000002245 particle Substances 0.000 description 1
- 230000009467 reduction Effects 0.000 description 1
Images
Classifications
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B64—AIRCRAFT; AVIATION; COSMONAUTICS
- B64G—COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
- B64G1/00—Cosmonautic vehicles
- B64G1/22—Parts of, or equipment specially adapted for fitting in or to, cosmonautic vehicles
- B64G1/24—Guiding or controlling apparatus, e.g. for attitude control
- B64G1/244—Spacecraft control systems
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
- G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
- G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
Abstract
The invention discloses a method for realizing spin stability and deployment optimization of a tethered formation by using a continuous propeller, belonging to the field of spacecraft guidance and control. The implementation method of the invention comprises the following steps: defining different reference coordinate systems to describe the motion of the rope formation system, and establishing a dynamic model of the spinning stable rope formation system by using a Lagrange equation; describing a finite time domain optimal control problem in a real-time domain, then converting the finite time domain optimal control problem into a Mayer form based on time domain mapping, giving control input and state variable constraints to a dynamic model of the spin-stable rope formation system, and constructing an optimal expansion model of the spin-stable rope formation system; the method is characterized in that a Legendre-Gauss discretization method is utilized, the final state and control input of the spin stable unfolding of the rope formation system are discretized on a series of discrete points, and the numerical solution is carried out on the dynamic process of the spin stable unfolding of the rope formation system through a Gaussian pseudo-spectrum method, so that parameters needing to be input can be reduced, and the calculation accuracy is improved.
Description
Technical Field
The invention relates to a method for realizing spin stability and deployment optimization of a tethered formation by using a continuous propeller, in particular to a method for searching an open-loop optimal track and guiding a subsatellite from an initial position to a final rotating state, and belongs to the field of spacecraft guidance and control.
Background
The tethered satellite system is a spacecraft system formed by connecting two or more satellites by a tether, has wide application prospects in the fields of atmospheric exploration, space environment exploration, space debris removal and the like, and can be used as a new structure for exploring space, acquiring resources in the space and widening human living space by human beings. For example, the space exploration platform is released from the main satellite to the atmosphere of the earth by using a tether, so that the near-earth atmosphere is explored. When the tethered satellite system runs at high speed, the tether cuts the terrestrial magnetic force line, generates current, and lifts or lowers the system orbit under the action of the terrestrial magnetic field without consuming fuel.
In recent years, the concept of building large flexible space systems using spin-stabilized tethered formations has attracted considerable attention. The spin-stabilized formation of ropes can be maintained in its configuration by spin-generated centrifugal forces, which allows the control of the space system with less fuel consumption than conventional space systems. Furthermore, by adjusting the length of the connecting ropes, the formation can also provide a long and variable base line, so that high-quality space observation can be performed. But the dynamic coupling and constraint control nonlinearity problems exist in the spinning stable unfolding process of the rope formation system.
And selecting a Lagrange method for modeling the tethered satellite system only considering the stretching condition. If the tethered convoy comprises three or more stars, the convoy operation can be difficult due to practical challenges, one of which is to deploy the convoy from an initial state to a final desired configuration. Due to coriolis acceleration and gravity gradients, the subsatellite experiences in-plane and out-of-plane vibrations, causing slack in the tether and even collisions between adjacent components. Overall, these challenges mainly come from the complexity of the formation dynamics; while many researchers have investigated the control problem of roping, spin-stable roping with a central body is less of a concern, making deployment of the roping more complex.
Disclosure of Invention
Aiming at the problem of the dynamic coupling and constraint control nonlinearity in the self-rotation stable unfolding process of a rope formation system, the defects of more parameters needing to be input and poor calculation accuracy in solving a specific nonlinear equation exist, the invention discloses a method for realizing self-rotation stable unfolding optimization of rope formation by using a continuous propeller, which aims to solve the technical problems that: the method is characterized in that a Legendre-Gauss discretization method is utilized, the final state and control input of the spin stable unfolding of the rope formation system are discretized on a series of discrete points, and the numerical solution is carried out on the dynamic process of the spin stable unfolding of the rope formation system through a Gaussian pseudo-spectrum method, so that the parameters needing to be input can be reduced, and the calculation accuracy is improved. The numerical solution can provide a high-precision control scheme for the rope formation system, so that the self-spinning stable unfolding control precision of the rope formation system is improved.
The purpose of the invention is realized by the following technical scheme.
The invention discloses a method for realizing spin stability unfolding optimization of a rope formation system by using a continuous propeller, which defines different reference coordinate systems to describe the motion of the rope formation system and establishes a dynamic model of the spin stability rope formation system by using a Lagrange equation. Describing a finite time domain optimal control problem in a real-time domain, then converting the finite time domain optimal control problem into a Mayer form based on time domain mapping, giving control input and state variable constraints to a dynamic model of the spin-stabilized tether formation system, and constructing an optimal expansion model of the spin-stabilized tether formation system. The method is characterized in that a Legendre-Gauss discretization method is utilized, the final state and control input of the spin stable unfolding of the rope formation system are discretized on a series of discrete points, and the numerical solution is carried out on the dynamic process of the spin stable unfolding of the rope formation system through a Gaussian pseudo-spectrum method, so that the parameters needing to be input can be reduced, and the calculation accuracy is improved.
The invention discloses a method for realizing spin stability and unfolding optimization of rope formation by using a continuous propeller, which comprises the following steps:
the method comprises the following steps: defining different reference coordinate systems to describe the motion of the rope formation system, and establishing a dynamic model of the spinning stable rope formation system by using a Lagrange equation;
step 1.1: defining an earth inertia coordinate system ECI, a local vertical local horizontal LVLH coordinate system and a body fixed connection coordinate system;
the earth inertial coordinate system ECI, expressed in xyz, is a non-rotating coordinate system; the x-axis and the y-axis are located on the equatorial plane, the x-axis coincides with the spring equinox, the Z-axis passes through the north pole of the earth, and the y-axis is determined by the right-hand rule. To be provided withOxoyozoThe local vertical local horizontal LVLH coordinate system represented is connected with the main star, and z thereofoThe axis is along the radial vector, x, from the spacecraft to the earth's centroidoThe axis being the vector y along the track speedoThe axes are determined by the right hand rule. Is shown as OxbybzbBody of (2) fixedly connecting coordinatesIs used to define a dynamic variable whose origin is fixed at the centroid, x, of the principal starbThe axis is called local radial and originates from the origin to the tether attachment point, ybThe axis being aligned with the axis of rotation of the system, zbThe axes are determined by the right hand rule.
Step 1.2: calculating the total kinetic energy of the rope formation system;
when the formation is symmetric along the rotation axis, within the local vertical local horizontal coordinate system, it is obtained:
wherein: rhoi＝[ρxρyρz]TA position vector representing the ith subsatellite relative to the local vertical local horizontal coordinate system is further represented as:
wherein: r is the radius of the main star, θiis the spin angle of the main star αiIs measured from the connecting ropes to the local radial vibration angle,/iIndicating the length of the connecting tether. In addition, viExpressed as the subsatellite velocity in the earth's inertial coordinate system, is calculated by equation (3):
vi＝vo+vci(3)
wherein: v. ofoRepresenting the velocity, v, of the system centroid in the earth's inertial frameciRepresenting the velocity of the subsatellite relative to the primary satellite in the earth's inertial frame. The speed v of the system mass center in the earth inertia coordinate system is due to the formation in a circular orbitoCalculated from equation (4):
v0＝Ω×R (4)
wherein [ omega ], [ alpha ], [0，-Ω，0]TIs the track velocity vector and R is the track position vector at the center of the formation. In addition, vciCalculated from equation (5):
the formula (2) can be substituted for the formula (5):
finally, the total kinetic energy of the rope formation system is obtained by formula (7):
wherein:
Tci＝mi(vo+vci)·(vo+vci) (9)
wherein: j. the design is a squarecRepresenting the moment of inertia, m, of the main star relative to the axis of rotationcAnd miRepresenting the masses of the main star and the sub-star, respectively. The second term on the right of equation (7) can be extended to:
obtained by the formula (1):
substituting formula (3), formula (4) and formula (11) into formula (10) then yields:
step 1.3: calculating the total potential energy of the rope formation system;
neglecting earth oblateness disturbance, formulating the gravitational potential energy of the rope formation system as:
wherein: mu.seIs a constant gravity coefficient, and (Rc + ρ i) is then-1The terms are extended to taylor series and the higher order terms are ignored:
where ρ isi＝||ρiL. Substituting formula (14) for formula (13) ultimately yields:
step 1.4: and establishing a dynamic model of the spin-stable rope formation system by utilizing a Lagrange equation, and expressing the dynamic model of the spin-stable rope formation system in a state space form.
Lagrange's equation describes the formation's motion according to kinetic and potential energy conditions:
wherein: q. q.sjIs a generalized coordinate, QjRepresenting a generalized force acting on the system. By selecting the previously defined independent variables as generalized coordinates and substituting the kinetic and potential energy terms into equation (16), the kinetic model of the spin-stable tether formation system as shown in equations (17) to (19) is obtained:
wherein:f＝[uθ，uαi，uli]Trepresents the active control input vector, and f ═ ud θ, ud α i, udli]TRepresenting the perturbations in relation to the gravity gradient and the centrifugal force, each perturbation component being represented as:
the primary star is generally known as the particle in the prior art, and thus the dynamic problem treatment of the dynamic coupling between the primary star and the tether is limited. Due to the complexity of dynamics and perturbations, it is difficult to design an analytical control scheme for the development of formation. Therefore, only an approximation of the formation dynamics is usually available. When the main star continuously rotates under the complete active control, the tether formation system is decoupled, and the dynamic decoupling analysis of each subsatellite is realized. Thus, for each subsatellite, the motions of pitch angle and tether length are described independently by equations (23), (24), respectively:
wherein:
is the constant rotation rate of the primary star. Equation (23) describes the in-plane vibration of the tether formation system, and equation (24) indicates the change in length of the tether. Since the kinetic form of each subsatellite is the same, without loss of generality, the index i of any symbol is ignored. During the unwinding process, the rotation rate of the formation is always greater than the track angular velocity; thus, formula (A)20) The perturbation term into equation (22) can be ignored. Finally, a state vector is defined
The dynamic model of the spin-stable tether formation system is expressed by a state space formula as follows:
wherein: u ═ fα，fl]TRepresenting the control input, the vector function denoted f (x, u) is:
step two: describing a finite time domain optimal control problem in a real-time domain, and then converting the finite time domain optimal control problem into a Mayer form based on time domain mapping; and C, setting control input and state variable constraints on the dynamic model of the spin-stable rope formation system established in the step I, and constructing an optimal expansion model of the spin-stable rope formation system.
Step 2.1: describing a finite time domain optimal control problem in a real-time domain;
the spin-stable tether formation system in the real-time domain is represented as:
wherein t is an element [ t ]0，tf]X (t) is the system state and u (t) is the control input. The cost function is defined as:
the finite time domain optimal control problem is to determine that the state control minimizes the cost function in equation (28) and obeys the dynamics and given constraints in equation (27).
x(t0)＝x0x(tf)＝xf(29)
Ψ(x)≥0 t∈[to，tf](30)
Θ(u)≥0 t∈[t0，tf](31)
Wherein: equation (29) is a boundary condition, and equations (30) and (31) are path constraints of the state and control inputs.
A constrained optimal control problem is solved in which the cost function appears in the form of an accompanying variable. The optimal control problem is in practice considered to be a two-point boundary problem, with the augmented Hamiltonian amount typically defined in terms of system state, control inputs and accompanying variables. The vanishing gradient of the Hamiltonian provides necessary conditions for Pontryagin optimization, and finally an optimal solution can be obtained according to the necessary conditions.
Step 2.2: the finite time domain optimal control problem described in step 2.1 is converted into a Mayer form based on time domain mapping.
And under the framework of a Gaussian pseudo-spectral method, the state variable and the control variable of the optimal control problem are approximately solved at a Legendre Gauss point by utilizing a Lagrange polynomial. The state derivatives of the state variables are then calculated at Legendre Gauss points through a differential matrix, and at the same time, the continuous constraints are converted into discrete algebraic constraints. Since Legendre Gauss point is [ -1, 1 [)]So that the prototype of the optimal control problem is modeled by mapping the real-time interval to [ -1, 1 [ -1 [ ]]the domain reformulates a Mayer type, for the real-time interval t e [ t ∈0，tf]The domain mapping is achieved by the following transformation:
from equation (32), one can derive:
and:
after time domain mapping, the first derivative of the state is calculated:
by formulas (34) and (35), the kinetic rewrite is:
the optimal control problem in the form of Mayer can then be expressed as follows: the state control pairs are found in the-1, 1 domain, minimizing the cost function.
Subject to the dynamic constraints of equation (36) and the boundary conditions:
x(τ＝-1)＝x0x(τ＝-1)＝xf(38)
and (3) path constraint:
Ψ(x(τ))≥0 τ∈[-1，1](39)
Θ(u(τ))≥0 τ∈[-1，1](40)
step 2.3: converting the finite time domain optimal control problem described in the step 2.1 and the step 2.2 into a Mayer form based on time domain mapping; and C, setting control input and state variable constraints on the dynamic model of the spin-stable rope formation system established in the step I, and constructing an optimal expansion model of the spin-stable rope formation system.
Initial deployment conditions include initial vibration angle and rate and tether length and rate:
after the rope formation system is successfully deployed, the final state is expected to be reached:
wherein alpha isfAnd lfRepresenting the final vibration angle and tether length. During deployment, the tether tension should be limited to:
wherein:
and
respectively representing a lower boundary and an upper boundary; the maximum allowable amplitude and the minimum allowable amplitude of the connecting ropes are determined;
a continuous pusher is utilized to generate a control force for deployment of the formation. Since the tether provides only resistance, the provision of a longitudinal thruster produces a positive force. The longitudinal control force can thus be extended to a range from negative to positive, the total longitudinal control force along the roping being expressed as:
fl＝ftether+fthruster1(44)
due to the amplitude limitation of the propeller, the forces generated by the longitudinal propeller are constrained as follows:
wherein: for a tangential thruster, the limits of the continuous control force are as follows:
to enable real-time monitoring, the motion of the subsatellite is checked using a camera. During unfolding, the subsatellite must be within the FOV of the camera's field of view. Therefore, the equivalent constraint of the vibration angle satisfies:
-αmax≤α≤αmax(47)
wherein alpha ismaxIs determined by the FOV range of the surveillance cameraAnd (4) determining a vibration angle boundary.
Finally, the cost function for minimizing power consumption is expressed as:
equation (48) obeys the following constraints and boundary conditions: the dynamic constraint of equation (27), the boundary conditions of equations (41) and (42), the input constraints of equations (43), (45), and (46), and the path constraint of equation (47).
The cost function for minimizing the power consumption is the optimal expansion model of the constructed spin-stable tether formation system.
Step three: the method is characterized in that a Legendre-Gauss discretization method is utilized, the final state and control input of the spin stable unfolding of the rope formation system are discretized on a series of discrete points, and the numerical solution is carried out on the dynamic process of the spin stable unfolding of the rope formation system through a Gaussian pseudo-spectrum method, so that the parameters needing to be input can be reduced, and the calculation accuracy is improved. The numerical solution can provide an analytic and high-precision control scheme for the rope formation system, so that the spinning stability unfolding control precision of the rope formation system is improved.
Using the Lagrangian interpolation polynomial, the final state and control inputs are approximated at the Legendre Gauss point to produce a polynomial of degree N:
wherein: k is the number of Legendre Gauss points and is a Lagrangian interpolation polynomial.
As shown in equations (49) and (50), Lagrangian interpolation can ensure x (τ)i)＝X(τi) And u (τ)i)＝U(τi) Wherein the value of the non-Legendre Gauss point is only approximately equal to the true value. The differentiation of equation (49) is:
wherein: differential matrix DkiThe determination is as follows:
wherein K is 1,2, K and PK(τ) is a Legendre polynomial of order K, expressed as:
in the formula (27) in which the formula (55) is substituted, the dynamics of the rope formation system are rewritten as follows:
namely, the final state and the control input of the spinning stable unfolding of the rope formation system are dispersed on a series of discrete points by utilizing a Legendre-Gauss discretization method.
Furthermore, the final state is obtained by gaussian multiplication:
wherein: omegakAnd DKiIs determined only by the number of Legendre Gauss points and is calculated offline before the next step of numerical propagation. And finally, based on a Gaussian pseudo-spectrum method, converting the prototype optimal control problem into a nonlinear programming problem in the following form:
namely, the numerical solution of the systematic spin stability unfolding dynamic process of the rope formation system is realized by the Gaussian pseudo-spectrum method through the formulas (57) and (58), so that the parameters needing to be input can be reduced, and the calculation precision is improved. The numerical solution can provide a high-precision control scheme for the rope formation system, so that the self-spinning stable unfolding control precision of the rope formation system is improved.
Has the advantages that:
1. the invention discloses a method for realizing spin stability and unfolding optimization of a rope formation system by using a continuous propeller, which takes the spin of a main star into consideration in a circular orbit, provides a dynamic model for the spin unfolding of a complex rope formation system taking the rotation of the main star into consideration, and improves the modeling precision of the complex rope formation system with the rotation of the main star.
2. The invention discloses a method for realizing the self-spinning stable unfolding optimization of a tethered formation by using a continuous propeller, which is characterized in that the optimal control problem of a limited time domain is described in a general form in the time domain, then the optimal control problem is converted into a Mayer form based on time domain mapping, and then the optimal unfolding problem is constructed by using formation dynamics and operation constraints of control input and state variables, so that the control force for the formation unfolding is generated by using the continuous propeller, and meanwhile, the longitudinal control force can be expanded to a range from negative to positive.
3. The invention discloses a method for realizing spin stability unfolding optimization of a rope formation system by using a continuous propeller, which is characterized in that the final state and control input of the spin stability unfolding of the rope formation system are scattered on a series of discrete points by using a Legendre-Gauss discretization method, and the dynamic process of the spin stability unfolding of the rope formation system is numerically solved by using a Gaussian pseudo-spectrum method, so that the parameters needing to be input can be reduced, and the calculation accuracy is improved. The numerical solution can provide a high-precision control scheme for the rope formation system, so that the self-spinning stable unfolding control precision of the rope formation system is improved.
Description of the drawings:
FIG. 1 is a flow chart of a method for optimizing the self-spinning stable deployment of a formation of ropes by using a continuous thruster according to the present invention;
FIG. 2 is a schematic diagram of tethered satellite formation according to the present invention;
FIG. 3 is a schematic diagram of the step two expansion constraint of the present invention;
FIG. 4 is a schematic view of the vibration angle and angular velocity according to embodiment A of the present invention;
FIG. 5 is a schematic representation of tether length and deployment rate according to example A of the present invention;
FIG. 6 is a schematic diagram of an unfolding path of a subsatellite in a local vertical local horizontal coordinate system and a body-fixed coordinate system in the embodiment A of the present invention;
FIG. 7 is a schematic diagram of an optimal control input according to embodiment A of the present invention;
FIG. 8 is a schematic view of the vibration angle and angular velocity according to embodiment B of the present invention;
FIG. 9 is a schematic representation of tether length and deployment rate according to example B of the present invention;
FIG. 10 is a schematic diagram of the unfolding path of the subsatellite in the local vertical local horizontal coordinate system and the body fastening coordinate system in embodiment B of the present invention;
FIG. 11 is a schematic diagram of an optimal control input according to embodiment B of the present invention;
FIG. 12 is an expanded graph of stateful components with different rotation rates in an embodiment of the present invention;
FIG. 13 is a graph of control curves with different rotation rates for an embodiment of the present invention;
FIG. 14 is a graph illustrating the cost function of the thruster and the sensitivity of the maximum amplitude to the spin rate in an embodiment of the present invention;
FIG. 15 is a graph of the deployment of status components having different deployment times in an embodiment of the present invention;
FIG. 16 is a control graph with different deployment times for an embodiment of the present invention;
FIG. 17 is a graphical representation of propeller cost function and maximum amplitude versus deployment time sensitivity results in an embodiment of the present invention.
Detailed Description
To better illustrate the objects and advantages of the present invention, the following detailed description of the embodiments of the present invention is provided in conjunction with the accompanying drawings.
Examples a and B:
in order to verify the feasibility of the optimization method for realizing the stable self-spinning unfolding of the tether formation by using the continuous thruster, a tether satellite numerical model is firstly constructed. The basic system parameters are shown in table 1. For ease of comparison, two typical deployment scenarios were considered in the simulation. The first relates to tangential and longitudinal thrusters, while the second relates only to tangential thrusters. To avoid the convergence problem caused by the inequality constraint, the algorithm of the present embodiment iteratively determines the number/width of intervals and the polynomial degree of each interval; thus, the number of nodes and the degree of the polynomial are continually updated until a specified tolerance is met, and the pseudo-linearization propagation is designed to be if and only if all Legendre Gauss points meet | u (k +1) -u (k) | ≦ 10-3And then terminates. The parameters for both cases are shown in table 2.
TABLE 1 System parameters
Table 2 parameters of the examples
As shown in fig. 1, the present embodiment discloses a method for optimizing the spin-stable deployment of a rope formation by using a continuous thruster, which includes the following steps:
the method comprises the following steps: defining different reference coordinate systems to describe the motion of the rope formation system, and establishing a dynamic model of the spinning stable rope formation system by using a Lagrange equation;
defining a state vector
The dynamic model of the spin-stable tether formation system is expressed by a state space formula as follows:
wherein u ═ f α, fl]TRepresenting the control input, the vector function denoted f (x, u) is:
where r is the radius of the main star, α is the vibration angle measured from the connecting tether to the local radial direction, l represents the length of the connecting tether, Ω is the orbital velocity vector, and ω is the constant rotation rate of the main star.
So the vector function represented by f (x, u) is rewritten as:
step two: describing a finite time domain optimal control problem in a real-time domain, and then converting the finite time domain optimal control problem into a Mayer form based on time domain mapping; and C, setting control input and state variable constraints on the dynamic model of the spin-stable rope formation system established in the step I, and constructing an optimal expansion model of the spin-stable rope formation system.
The cost function for minimizing power consumption is expressed as:
in summary, equation (48) obeys the following constraints and boundary conditions:
-αmax≤α≤αmax(11)
wherein t ∈ [0, 120]x (t) is the system state, u (t) is the control input, αfAnd lfIndicating the final vibration angle and tether length,
and
respectively representing a lower and an upper boundary, alphamaxPi/3 rad is the vibration angle boundary determined by the surveillance camera FOV range.
Step three: the method is characterized in that a Legendre-Gauss discretization method is utilized, the final state and control input of the spin stable unfolding of the rope formation system are discretized on a series of discrete points, and the numerical solution is carried out on the dynamic process of the spin stable unfolding of the rope formation system through a Gaussian pseudo-spectrum method, so that the parameters needing to be input can be reduced, and the calculation accuracy is improved. The numerical solution can provide an analytic and high-precision control scheme for the rope formation system, so that the spinning stability unfolding control precision of the rope formation system is improved.
Using the Lagrangian interpolation polynomial, the final state and control inputs are approximated at the Legendre Gauss point to produce a polynomial of degree N:
wherein: k is the number of Legendre Gauss points and is a Lagrangian interpolation polynomial.
As shown in equations (12) and (13), lagrange interpolation can ensure x (τ)i)＝X(τi) And u (τ)i)＝U(τi) Wherein the value of the non-Legendre Gauss point is only approximately equal to the true value. The differentiation of equation (12) is:
wherein: differential matrix DkiThe determination is as follows:
wherein K is 1,2, K and PK(τ) is a Legendre polynomial of order K, expressed as:
in the formula (5) in which the formula (18) is substituted, the dynamics of the rope formation system are rewritten as follows:
namely, the final state and the control input of the spinning stable unfolding of the rope formation system are dispersed on a series of discrete points by utilizing a Legendre-Gauss discretization method.
Wherein: omegakAnd DKiIs determined only by the number of Legendre Gauss points and is calculated offline before the next step of numerical propagation. Based on the Gaussian pseudo-spectrum method, the prototype optimal control problem is converted into a nonlinear programming problem.
As shown in fig. 3, the vibration angle changes smoothly from zero, almost decreasing to-1 rad, and remains negative throughout, indicating that the subsatellite is always located behind the local radial during the rotational development. In addition, due to the constraints of camera surveillance, the amplitude of the vibration angle is limited by the boundary of- π/3rad, and after 40s, the vibration angle gradually increases to zero. The angular rate of vibration starts with a rapid oscillation at the initial stage. This rapid oscillation coincides well with the initial rapidly decreasing oscillation angle, but after the initial short-term oscillation, the oscillation angular velocity changes slowly and eventually approaches zero.
As shown in fig. 4, the tether length and the deployment rate change smoothly during deployment, the tether deployment rate remaining always positive, indicating that the tether length is always increasing and eventually reaches a length of 500 meters, while the tether deployment rate reaches a maximum at t-60 s and approaches zero at the final end.
Figure 5 shows the subsatellite trajectory in a local vertical local horizontal coordinate system and a body-attached coordinate system. The results show that no significant rail vibration occurs during the entire unwinding period, which means that the convoy remains good and eventually a stable configuration on the rail is achieved. The results can support relative motion analysis of the sub-stars relative to the main star.
Figure 6 plots the control input profile including the components of tether tension, tangential and longitudinal thruster forces. The result shows that all input components change within the corresponding limit range before t is 20s, the tangential thruster always generates negative force, and the deviation angle of the subsatellite is adjusted by small oscillation. Before t 60s, the longitudinal thruster provides positive force to radially deploy the tether, but after t 50s, the longitudinal thruster remains inactive until deployment is complete. As shown in fig. 6, the tether tension is maintained at 1N up to 73s and then gradually increased to 3.6N at the end of deployment.
Fig. 7 shows that the vibration angle and angular rate undergo initial oscillation, but the oscillation range of the two variables is greater than that of example a, which results in significant oscillation of the subsatellite trajectories in the local vertical local horizontal and body-attached coordinate system, as shown in fig. 9. The change in tether length and deployment rate is shown in figure 8. The results show that the tether length continues to increase throughout the deployment process, eventually reaching a length of 500 m. The spread rate reached its maximum at t 56s, but compared to the results of example a, the tether spread rate appeared to oscillate significantly before t 20 s. The oscillation can be seen in both the local vertical local horizontal and body-attached coordinate systems.
In figure 10 it is plotted that the amplitude of the tangential thruster is always kept at zero, while the amplitude of the longitudinal thruster varies within the range of-5, 8N. The roping tension remains 1N up to 82s and then gradually increases to a maximum of 4N. It reached 3.6N at the end of the expansion, consistent with the results of example A. More importantly, fig. 10 shows that by deactivating the tangential thrusters for deployment during the deployment phase, the maximum amplitude of the active thrusters should be increased to ensure that the solution is viable, and that increased power consumption is required for successful deployment.
The optimal deployment problem can be solved by a deployment time of 120s, as shown in fig. 11, where all state variables have the same trend of change during deployment. The vibration angle variation is smooth but always limited by the FOV constraints and the initial oscillation occurs almost simultaneously in all cases. It is noteworthy that the peak of the initial oscillation of the in-plane vibration angular rate and the maximum of the tether deployment rate show typical sensitivity to changes in rotation rate. The results show that increasing the rotation rate decreases the peak of the angular rate of vibration while increasing the maximum tether deployment rate.
The result of the optimal control input is shown in fig. 12. For all cases, no significant oscillation of the tangential thrust and tether tension occurs, but the maximum amplitude of the two input components increases with increasing spin rate of the main star. The reasoning result obtained by analyzing the bottom curve has the characteristics of vividness: if the rotation speed is omega 2 pi/80 rad/s, the longitudinal thrust is always kept at 1N; at higher rotation rates, however, overshoot will occur at the beginning of the unrolling. The figure also shows that a lower rate of rotation will correlate to a larger peak in longitudinal thrust.
To demonstrate the sensitivity of the cost function and the maximum thrust amplitude to the rotation speed, the rotation period was increased from 60s to 160s with a time interval of 10 s. The results of the different spin rates are shown in fig. 13, demonstrating that by reducing the spin rate, the maximum amplitudes of the cost function and the tangential thrust (negative) both decrease monotonically, but when the spin period is greater than 120s, the decrease becomes insignificant. The curve shows the sensitivity of the maximum longitudinal thrust to the rotation rate, whereas the phenomenon of increased sensitivity occurs only when the spin period is longer than 90s, compared to the result of the tangential thrust, while the maximum value remains almost unchanged when the rotation period is shorter than 90 s.
By selecting the deployment times of 80s, 100s and 120s, a spin cycle result of 80s is obtained, as shown in FIG. 14. The graph shows that with each increase in deployment time, all time histories of the variables change with the same trend, and that neither the vibration angular rate nor the peak values of the tether deployment rate change. As shown in fig. 15, the tangential thrust and the tether tension have a similar trend even at different deployment times, whereas for the longitudinal thrust, the overshoot occurs at a deployment time of 80s, but for other cases it remains almost constant.
The cost function and the sensitivity of the maximum thrust amplitude to the deployment time were also analyzed in a series of simulations, where the deployment time was increased from 50s to 140s with a time interval of 10 s. As shown in fig. 16, all results show that the amplitude decreases as the deployment time increases. It should be noted, however, that when the deployment time exceeds 90s, the reduction in all results becomes insignificant.
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (3)
1. A method for realizing spin stability unfolding optimization of rope formation by using a continuous propeller is characterized by comprising the following steps: comprises the following steps of (a) carrying out,
the method comprises the following steps: defining a reference coordinate system to describe the motion of the rope formation system carrying the continuous propeller, and establishing a dynamic model of the spinning stable rope formation system by using a Lagrange equation;
step two: describing a finite time domain optimal control problem in a real-time domain, and then converting the finite time domain optimal control problem into a Mayer form based on time domain mapping; setting control input and state variable constraints on the dynamic model of the spin-stable rope formation system established in the step one, and constructing an optimal expansion model of the spin-stable portable continuous propeller rope formation system;
step three: dispersing the final spin stable unfolding state and control input of the rope formation system on a series of discrete points by using a Legendre-Gauss discretization method, and performing numerical solution on the dynamic process of the spin stable unfolding of the rope formation system by using a Gaussian pseudo-spectrum method, so that the parameters needing to be input can be reduced, and the calculation precision is improved; the numerical solution can provide an analytic and high-precision control scheme for the rope formation system, so that the self-rotation stability unfolding control precision of the rope formation system with the continuous propeller is improved;
the specific implementation method of the step one is that,
step 1.1: defining an earth inertia coordinate system ECI, a local vertical local horizontal LVLH coordinate system and a body fixed connection coordinate system;
the earth inertial coordinate system ECI, expressed in xyz, is a non-rotating coordinate system; the x axis and the y axis are positioned on an equatorial plane, the x axis is coincided with a spring minute point, the z axis passes through the north pole of the earth, and the y axis is determined by a right-hand rule; with OxoyozoThe local vertical local horizontal LVLH coordinate system represented is connected with the main star, and z thereofoThe axis is along the radial vector, x, from the spacecraft to the earth's centroidoThe axis being the vector y along the track speedoThe axis is determined by the right hand rule; is shown as OxbybzbThe body of (2) is fixedly connected with a coordinate system for defining a dynamic variable, and the origin of the dynamic variable is fixed at the mass center x of the main starbThe axis is called local radial and originates from the origin to the tether attachment point, ybThe axis being aligned with the axis of rotation of the system, zbThe axis is determined by the right hand rule;
step 1.2: calculating the total kinetic energy of the rope formation system;
when the formation is symmetric along the rotation axis, within the local vertical local horizontal coordinate system, it is obtained:
wherein: m isiRepresents the ith subsatellite mass, ρi＝[ρxρyρz]TA position vector representing the ith subsatellite relative to the local vertical local horizontal coordinate system is further represented as:
wherein: r is the radius of the main star, θiis the spin angle of the main star, αiIs measured from the connecting ropes to the local radial vibration angle,/iIndicating the length of the connecting roping; in addition, viExpressed as the subsatellite velocity in the earth's inertial coordinate system, is calculated by equation (3):
vi＝v0+vci(3)
wherein: v. ofoRepresenting the velocity, v, of the system centroid in the earth's inertial frameciRepresenting the velocity of the subsatellite relative to the main satellite in the earth inertial coordinate system; the speed v of the system mass center in the earth inertia coordinate system is due to the formation in a circular orbitoCalculated from equation (4):
v0＝Ω×R (4)
where Ω is the track velocity vector, and R is the track position vector of the formation center; in addition, vciCalculated from equation (5):
the formula (2) can be substituted for the formula (5):
finally, the total kinetic energy of the rope formation system is obtained by formula (7):
wherein: t iscAnd TciCalculated from equation (8) and equation (9):
Tci＝mi(vo+vci)·(vo+vci) (9)
wherein: j. the design is a squarecRepresenting the moment of inertia, m, of the main star relative to the axis of rotationcAnd miRespectively representing the mass of the main star and the sub-star; the second term on the right of equation (7) can be extended to:
obtained by the formula (1):
substituting formula (3), formula (4) and formula (11) into formula (10) then yields:
step 1.3: calculating the total potential energy of the rope formation system;
neglecting earth oblateness disturbance, formulating the gravitational potential energy of the rope formation system as:
wherein: mu.seIs a constant gravity coefficient, which will be (R + ρ i)-1The terms are extended to taylor series and the higher order terms are ignored:
where ρ isi＝||ρiI, calculating the norm of the position vector; substituting formula (14) for formula (13) ultimately yields:
step 1.4: establishing a dynamic model of the spin-stable rope formation system by using a Lagrange equation, and expressing the dynamic model of the spin-stable rope formation system in a state space form;
lagrange's equation describes the formation's motion according to kinetic and potential energy conditions:
wherein: q. q.sjIs a generalized coordinate, QjRepresenting a generalized force acting on the system; by selecting the previously defined independent variables as generalized coordinates and substituting the kinetic and potential energy terms into equation (16), the kinetic model of the spin-stable tether formation system is obtained as shown in equations (17) to (19):
wherein f isθ，fαiAnd fliRepresents an active control input vector, and fdθ，fdαiAnd fdliRepresenting the perturbations in relation to the gravity gradient and the centrifugal force, each perturbation component being represented as:
when the main star continuously rotates under the complete active control, the tether formation system is decoupled, and the dynamic decoupling analysis of each subsatellite is realized; thus, for each subsatellite, the motions of pitch angle and tether length are described independently by equations (23), (24), respectively:
wherein:
is the constant rotation rate of the primary star; equation (23) describes the in-plane vibration of the tether formation system, equation (24) indicates the change in length of the tether; since the kinetic form of each subsatellite is the same, without loss of generality, the subscript i of any symbol is ignored; during the unwinding process, the rotation rate of the formation is always greater than the track angular velocity; therefore, the disturbance terms in equations (20) to (22) can be ignored; finally, a state vector is defined
The dynamic model of the spin-stable tether formation system is expressed by a state space formula as follows:
wherein u ═ f α, fl]TRepresenting the control input, the vector function denoted f (x, u) is:
2. the method for optimizing the self-spinning stable deployment of the formation of the ropes by using the continuous thruster as claimed in claim 1, wherein: the concrete implementation method of the second step is that,
step 2.1: describing a finite time domain optimal control problem in a real-time domain;
the spin-stable tether formation system in the real-time domain is represented as:
wherein t is an element [ t ]0，tf]X (t) is the system state, u (t) is the control input; the cost function is defined as:
the finite time domain optimal control problem is to determine that the state control minimizes the cost function in equation (28) and obeys the dynamics and given constraints in equation (27);
x(t0)＝x0x(tf)＝xf(29)
Ψ(x)≥0 t∈[to，tf](30)
Θ(u)≥0 t∈[t0，tf](31)
wherein: equation (29) is a boundary condition, and equations (30) and (31) are path constraints for state and control inputs;
solving a constrained optimal control problem, wherein the cost function appears in the form of an adjoint variable; the optimal control problem is actually considered as a two-point boundary problem, and the augmented Hamiltonian is usually defined according to the system state, control input and accompanying variables; the vanishing gradient of the Hamilton quantity provides necessary conditions for Pontryagin optimization, and finally an optimal solution can be obtained according to the necessary conditions;
step 2.2: converting the limited time domain optimal control problem described in the step 2.1 into a Mayer form based on time domain mapping;
in the framework of a Gaussian pseudo-spectral method, a Lagrange polynomial is used for approximately solving the state variable and the control variable of the optimal control problem at a Legendre Gauss point; then calculating the state derivative of the state variable at a Legendre Gauss point through a differential matrix, and simultaneously converting continuous constraint into discrete algebraic constraint; since Legendre Gauss point is [ -1, 1 [)]So that the prototype of the optimal control problem is modeled by mapping the real-time interval to [ -1, 1 [ -1 [ ]]the domain is used for reforming the Mayer type, and the real-time interval t is from the [ t ]0，tf]The domain mapping is achieved by the following transformation:
from equation (32), one can derive:
and:
after time domain mapping, the first derivative of the state is calculated:
by formulas (34) and (35), the kinetic rewrite is:
the optimal control problem in the form of Mayer can then be expressed as follows: find the state control pair in the [ -1, 1] domain, minimize the cost function;
subject to the dynamic constraints of equation (36) and the boundary conditions:
x(τ＝-1)＝x0x(τ＝-1)＝xf(38)
and (3) path constraint:
Ψ(x(τ))≥0 τ∈[-1，1](39)
Θ(u(τ))≥0 τ∈[-1，1](40)
step 2.3: converting the finite time domain optimal control problem described in the step 2.1 and the step 2.2 into a Mayer form based on time domain mapping; setting control input and state variable constraints on the dynamic model of the spin-stable rope formation system established in the step one, and constructing an optimal expansion model of the spin-stable rope formation system;
initial deployment conditions include initial vibration angle and rate and tether length and rate:
after the rope formation system is successfully deployed, the final state is expected to be reached:
wherein alpha isfAnd lfRepresenting the final vibration angle and tether length; during deployment, the tether tension should be limited to:
wherein:
and
respectively representing a lower boundary and an upper boundary; the maximum allowable amplitude and the minimum allowable amplitude of the connecting ropes are determined;
utilizing the continuous propeller to generate a control force for the formation to be unfolded; since the tether provides only resistance, the provision of a longitudinal thruster generates a positive force; the longitudinal control force can thus be extended to a range from negative to positive, the total longitudinal control force along the roping being expressed as:
fl＝ftether+fthruster1(44)
due to the amplitude limitation of the propeller, the forces generated by the longitudinal propeller are constrained as follows:
wherein:
represents the maximum thrust provided by the longitudinal thrusters; for tangential pushForce device, continuously controlling force limits as follows:
in order to realize real-time monitoring, a camera is used for checking the motion of the subsatellite; during unfolding, the subsatellite must be within the FOV of the camera; therefore, the equivalent constraint of the vibration angle satisfies:
-αmax≤α≤αmax(47)
wherein alpha ismaxIs the vibration angle boundary determined by the surveillance camera FOV range;
finally, the cost function for minimizing power consumption is expressed as:
equation (48) obeys the following constraints and boundary conditions: the dynamic constraint of equation (27), the boundary conditions of equations (41) and (42), the input constraints of equations (43), (45), and (46), and the path constraint of equation (47);
the cost function for minimizing the power consumption is the optimal expansion model of the constructed spin-stable tether formation system.
3. The method for optimizing the self-spinning stable deployment of the formation of the ropes by using the continuous thruster as claimed in claim 2, wherein: the third step is realized by the concrete method that,
using the Lagrangian interpolation polynomial, the final state and control inputs are approximated at the Legendre Gauss point to produce a polynomial of degree N:
wherein: k is LegThe number of endre Gauss points, X (τ) and U (τ) are lagrange interpolation polynomials generated approximately at the LegendreGauss point for the final state and control inputs; polynomial L satisfying Lagrange interpolationi(τ) and
can be expressed as:
as shown in equations (49) and (50), Lagrangian interpolation can ensure x (τ)i)＝X(τi) And u (τ)i)＝U(τi) Wherein the value of the non-Legendre Gauss point is only approximately equal to the true value; the differentiation of equation (49) is:
wherein: differential matrix DkiThe determination is as follows:
wherein K is 1,2, …, K, PK(τ) is a Legendre polynomial of order K, expressed as:
in the formula (27) in which the formula (55) is substituted, the dynamics of the rope formation system are rewritten as follows:
namely, the final state and control input of the spinning stable unfolding of the rope formation system are dispersed on a series of discrete points by utilizing a Legendre-Gauss discretization method;
furthermore, the final state is obtained by gaussian multiplication:
wherein: omegakAnd DKiIs determined only by the number of Legendre Gauss points and is calculated off-line before the next step of numerical propagation; and finally, based on a Gaussian pseudo-spectrum method, converting the prototype optimal control problem into a nonlinear programming problem in the following form:
the numerical solution of the systematic spin stability unfolding dynamic process of the rope formation system is realized by a Gaussian pseudo-spectrum method through formulas (57) and (58), so that the parameters needing to be input can be reduced, and the calculation precision is improved; the numerical solution can provide a high-precision control scheme for the rope formation system, so that the self-spinning stable unfolding control precision of the rope formation system is improved.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201811432334.0A CN110007681B (en) | 2018-11-28 | 2018-11-28 | Optimization method for realizing spin stability and unfolding of rope formation by using continuous propeller |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201811432334.0A CN110007681B (en) | 2018-11-28 | 2018-11-28 | Optimization method for realizing spin stability and unfolding of rope formation by using continuous propeller |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN110007681A CN110007681A (en) | 2019-07-12 |
| CN110007681B true CN110007681B (en) | 2020-06-26 |
Family
ID=67165048
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201811432334.0A Active CN110007681B (en) | 2018-11-28 | 2018-11-28 | Optimization method for realizing spin stability and unfolding of rope formation by using continuous propeller |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN110007681B (en) |
Families Citing this family (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN111552180A (en) * | 2020-04-30 | 2020-08-18 | 西北工业大学 | Tether system satellite system tether deployment oscillation suppression control method |
| CN113311863B (en) * | 2021-05-25 | 2022-07-08 | 南京航空航天大学 | Method for judging dynamic behavior stability of space linear type rope system observation system |
| CN113935176B (en) * | 2021-10-19 | 2022-05-10 | 北京理工大学 | Efficient dynamics modeling method for electrodynamic force rope derailing device |
| CN114253291B (en) * | 2021-12-15 | 2023-11-14 | 北京航空航天大学 | Spacecraft formation guidance method and system based on linear pseudo spectrum model predictive control |
| CN114895566B (en) * | 2022-05-24 | 2023-11-10 | 西北工业大学 | Chatter prediction method for milling process by adopting transmission matrix reduction technology |
Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US6135387A (en) * | 1997-09-17 | 2000-10-24 | Rheinmetall W&M Gmbh | Method for autonomous guidance of a spin-stabilized artillery projectile and autonomously guided artillery projectile for realizing this method |
| CN103901894A (en) * | 2014-04-14 | 2014-07-02 | 西北工业大学 | Spinning unfolding and folding optimum control method of dual-body star space tethered formation system |
| CN103914078A (en) * | 2014-04-10 | 2014-07-09 | 西北工业大学 | Two-body star space tethered formation system configuration retention coordination control method |
| CN105468011A (en) * | 2015-12-31 | 2016-04-06 | 北京理工大学 | Radiation open loop tethered satellite formation constant speed spinning deployment control method |
| CN108319136A (en) * | 2018-01-18 | 2018-07-24 | 西北工业大学 | A kind of tether tension control method based on model prediction |
-
2018
- 2018-11-28 CN CN201811432334.0A patent/CN110007681B/en active Active
Patent Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US6135387A (en) * | 1997-09-17 | 2000-10-24 | Rheinmetall W&M Gmbh | Method for autonomous guidance of a spin-stabilized artillery projectile and autonomously guided artillery projectile for realizing this method |
| CN103914078A (en) * | 2014-04-10 | 2014-07-09 | 西北工业大学 | Two-body star space tethered formation system configuration retention coordination control method |
| CN103901894A (en) * | 2014-04-14 | 2014-07-02 | 西北工业大学 | Spinning unfolding and folding optimum control method of dual-body star space tethered formation system |
| CN105468011A (en) * | 2015-12-31 | 2016-04-06 | 北京理工大学 | Radiation open loop tethered satellite formation constant speed spinning deployment control method |
| CN108319136A (en) * | 2018-01-18 | 2018-07-24 | 西北工业大学 | A kind of tether tension control method based on model prediction |
Non-Patent Citations (3)
| Title |
|---|
| Deployment strategies for planar multi-tethered satellite formation;ZhaiGuang,等;《Aerospace Science and Technology》;20171231;第71卷;第475-484页 * |
| 基于集群空间机器人的合作目标协同定位技术;翟光，等;《北京理工大学学报》;20141031;第34卷(第10期);第1034-1039页 * |
| 辐射开环绳系卫星编队自旋展开动力学与控制策略;苏飞，等;《航空学报》;20160925;第37卷(第9期);第2809-2819页 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN110007681A (en) | 2019-07-12 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN110007681B (en) | Optimization method for realizing spin stability and unfolding of rope formation by using continuous propeller | |
| CN104718508B (en) | The Three dimensional steerable of four rotor units | |
| CN104345738B (en) | A kind of rope system release stable control method and electric power rope system final stage leave the right or normal track stable control method | |
| Zhang et al. | Disturbance characteristics analysis of CMG due to imbalances and installation errors | |
| CN107505846B (en) | A kind of anti-interference attitude harmony verification device of Space Manipulator System and control method | |
| Kazim et al. | Disturbance-rejection-based optimized robust adaptive controllers for UAVs | |
| Guo et al. | Controlling a quadrotor carrying a cable-suspended load to pass through a window | |
| Wang et al. | Attitude and altitude controller design for quad-rotor type MAVs | |
| Liu et al. | Robust attitude control for quadrotors with input time delays | |
| CN104309822A (en) | Parameter optimization-based spacecraft single-pulse drop-shaped fly-around track hover control method | |
| CN114580224B (en) | Distributed pneumatic fusion track coupling attitude perturbation analysis method | |
| CN109613822A (en) | Control method is unfolded in spatial tether system based on Nonlinear Model Predictive Control | |
| Suprijono et al. | Direct inverse control based on neural network for unmanned small helicopter attitude and altitude control | |
| Siddhardha | Autonomous reduced-gravity enabling quadrotor test-bed: Design, modelling and flight test analysis | |
| Morgan et al. | Decentralized model predictive control of swarms of spacecraft using sequential convex programming | |
| Pao et al. | Verifying robust time-optimal commands for multimode flexible spacecraft | |
| Wang et al. | Optimal variable amplitudes input shaping control for slew maneuver of flexible spacecraft | |
| Gong et al. | Spin-stabilized solar sail for displaced solar orbits | |
| Kumar | Satellite attitude stabilization using fluid rings | |
| Hough et al. | Lyapunov vector fields for thrust-limited spacecraft docking with an elliptically-orbiting uncooperative tumbling target | |
| CN116520719A (en) | Method and system suitable for full physical simulation of attitude dynamics of flexible spacecraft | |
| Zhang et al. | Space target surveillance based on non-linear model predictive control | |
| Hassan et al. | Modeling, Trimming and Simulation of a Full Scale Helicopter | |
| Lyu et al. | Navigation and control scheme for space rendezvous and docking with maneuvering noncooperative target based on dynamic compensation | |
| Bestaoui et al. | Geometry of Translational Trajectories for an Autonomous Vehicle With Wind Effect |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |